Problem: In one kind of chemical reaction, unconverted reactants change into converted reactants. The fraction $a$ of reactants that have been converted increases at a rate proportional to the product of the fraction of converted reactants and the fraction of unconverted reactants. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{da}{dt}=ka(1-a)$ (Choice B) B $\dfrac{da}{dt}=\dfrac{ka}{1-a}$ (Choice C) C $\dfrac{da}{dt}=ka^2$ (Choice D) D $\dfrac{da}{dt}=\dfrac{k}{a(1-a)}$
Solution: The fraction of converted reactants is denoted by $a$. The fraction of unconverted reactants is $1-a$. The speed is the rate of change of the fraction of converted reactants, which is is represented by $a'(t)$, or $\dfrac{da}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the product of $a$ and $1-a$. In conclusion, the equation that describes this relationship is $\dfrac{da}{dt}=ka(1-a)$.